To prove "A iff B", you need to prove "B implies A" and "not B implies not A". You could do it in 2 separate sections, one starting with the statement x2 + 2x + 1 is even, the other starting with the opposite.

Proof:
It can be seen that $x^2+2x+1$ is even, as given in the problem. Therefore, $x^2+2x$ is odd, since an even number minus $1$ is odd. Then, we know that any integer multiplied by $2$ will be even, and an odd number minus an even number will be odd. Therefore, $x^2$ is odd. Since squares of odd numbers are the only odd squares, then $x$ has to be odd.